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If I have a circular membrane that is fixed by the edges and then I apply a uniform pressure to the membrane (let's say the membrane is the top of a circular chamber that has a rigid wall and flat bottom) by injecting compressed air, what shape will the membrane take? Parabolic? Hyperbolic? Is the shape very dependent if the membrane is parallel to the ground or perpendicular? Will it be different at the edges?

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  • $\begingroup$ -1. No reasearch effort. What have you done to find an answer? $\endgroup$ Sep 13, 2017 at 12:16

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Without pressure, as a result of gravity it is a catenary in any cross section, https://en.wikipedia.org/wiki/Catenary

It is assumed that the density of the membrane (and therefore its tension) is uniform. The catenary deviates according to the direction of gravity.

The force of pressure is perpendicular to the membrane at all points and would form a perfectly spherical shape if acting alone; depending upon the pressure magnitude and the weight of the membrane this may be an effect that dominates that of gravity.

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  • $\begingroup$ Is there a reference that explains why the membrane would have a spherical shape (assuming gravity was negligible)? $\endgroup$
    – Mike Wong
    Sep 12, 2017 at 21:26
  • $\begingroup$ physics.stackexchange.com/questions/137980/… $\endgroup$
    – JMLCarter
    Sep 12, 2017 at 21:53
  • $\begingroup$ Interesting, but does that justification hold in general or is it specific for cases with surface tension like a soap film (in my particular case, the membrane is solid). $\endgroup$
    – Mike Wong
    Sep 12, 2017 at 23:53
  • $\begingroup$ A (thin, idealized) membrane has no bending stiffness. Therefore, the pressure can only be resisted by tension in the membrane, since by symmetry the shear is zero (at least in your case of a circular boundary). It follows that the Gaussian curvature of the membrane shape is determined only by the pressure. For any constant pressure source, this means the shape will be a section of a sphere. If the bending stiffness for a (thick, solid) "membrane" is not negligible, you don't have a membrane but a thin or thick shell structure, and the general solution is much more complicated. $\endgroup$
    – alephzero
    Sep 13, 2017 at 3:11
  • $\begingroup$ Without pressure, as a result of gravity it is a catenary in any cross section. This does not seem correct to me. Do you have a proof? A rotated catenary would be denser at the bottom than at the top, whereas the chain has uniform density. So I think a membrane of uniform density would hang shallower than a catenary. ... Also, you are assuming that (like a chain) there is no stretching. $\endgroup$ Sep 13, 2017 at 12:27

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